Questions: Implicit Differentiation in Several Variables

Question 1

Given F(x, y) = x² + y³ − 5 = 0, which expression correctly gives dy/dx?

• A. dy/dx = −(2x)/(3y²)
• B. dy/dx = −(3y²)/(2x)
• C. dy/dx = (2x)/(3y²)
• D. dy/dx = (∂F/∂y)/(∂F/∂x)

Answer: (A)

The formula is dy/dx = −(∂F/∂x)/(∂F/∂y). Here ∂F/∂x = 2x and ∂F/∂y = 3y², so dy/dx = −2x/(3y²). Option B reverses numerator and denominator. Option C drops the negative sign. Option D inverts the ratio — a very common error, since the negative partial of x goes in the numerator, not denominator.

Question 2

When using the implicit differentiation formula dy/dx = −Fₓ/F_y, a student asks: 'Where does the negative sign come from?' The best answer is:

• A. It is a convention chosen to make the formula agree with known examples like the unit circle
• B. It arises because differentiating F(x, y(x)) = 0 with respect to x gives Fₓ + F_y(dy/dx) = 0, so dy/dx = −Fₓ/F_y
• C. It reflects the fact that as x increases, y must decrease to stay on the level set
• D. It comes from the negative slope of implicit curves, which always decrease

Answer: (B)

The negative sign is not a convention or a geometric observation — it is a direct algebraic consequence of the chain rule. Differentiating the constant F(x, y(x)) = 0 with respect to x produces two terms via the chain rule: the direct dependence ∂F/∂x, and the indirect dependence through y, which is (∂F/∂y)(dy/dx). Setting their sum to zero and solving gives the formula with its negative sign. Option C is geometrically true for some curves but is not the source of the sign.

Question 3

The formula dy/dx = −(∂F/∂x)/(∂F/∂y) is an independent rule specific to implicit functions, separate from the chain rule.

• T. True
• F. False

Answer: False

The formula is a direct application of the multivariable chain rule to the identity F(x, y(x)) = 0. It is not an independent rule; the chain rule is doing all the work. Understanding this derivation — rather than memorizing the formula — allows you to extend the technique to more variables, constrained systems, and eventually the implicit function theorem.

Question 4

If F(x, y) = 0 and ∂F/∂y = 0 at a point, the implicit differentiation formula dy/dx = −Fₓ/F_y breaks down at that point.

• T. True
• F. False

Answer: True

The condition ∂F/∂y ≠ 0 is essential: dividing by ∂F/∂y is only valid when it is nonzero. Geometrically, ∂F/∂y = 0 means the level set F = 0 has a vertical tangent or a singular point at that location — y may not be well-defined as a function of x near there. This is precisely the condition stated by the implicit function theorem for when the implicit function y(x) locally exists and is differentiable.

Question 5

Explain why the implicit differentiation formula dy/dx = −(∂F/∂x)/(∂F/∂y) requires the condition ∂F/∂y ≠ 0, and what its failure signals geometrically.

Model answer: The formula is derived by solving Fₓ + F_y(dy/dx) = 0 for dy/dx; dividing by F_y is only valid if F_y ≠ 0. If F_y = 0, the equation Fₓ = 0 either has no solution for dy/dx (if Fₓ ≠ 0) or leaves dy/dx undetermined. Geometrically, F_y = 0 means the level curve F = 0 has a vertical tangent — the curve turns back on itself so y is no longer a single-valued function of x. The implicit function theorem formalizes this: F_y ≠ 0 guarantees that y can be expressed as a smooth function of x near the point.

Students often apply the formula mechanically without checking the condition. At a vertical tangent, the tangent line is dy/dx → ∞, which the formula correctly signals by producing division by zero. At a cusp or self-intersection, F_y = 0 occurs at a singular point where the whole concept of 'the derivative' fails. The condition is not a technicality but a genuine check on whether differentiation makes sense.